Lecture 3: CAPM in practice

Lecture 3: CAPM in practice

Overview

1. The Markowitz model and active portfolio management. 2. A Note on Estimating

β

3. Using the single-index model and the CAPM in the passive portfolio problem

3.1 Usingβto get good covariance estimates 3.2 Usingβto get good expected-return estimates

4. The Black and Litterman Method.

,→ An Example:Global Asset Allocation

Using the Markowitz model

With a reasonable number of securities, the number of parameters that must be estimated is huge:

,→ For a portfolio of N (100) securities we need:

σi’s N 100

E(ri)’s N 100

Cov(ri,rj)’s 1₂N(N−1) 4950

Total 1

2N(N+3) 5150

,→ About how much data will we need when we have 500 securities? 1000 Securities?

Means and covariances are estimated witherror.

Small errors in mean or covariance estimates often lead to unreasonable weights.

Using the Markowitz model

A remedy for both of these problems is: 1. First, we use

I theCAPMto determine what market believes expected returns

should be

I asingle factor modelto calculate asset covariances

2. then we can combine our “views” with the CAPM-derived estimates to get portfolio weights.

The key input we will need for both of these is the set of asset

β

s. So, first, we must consider the problem of estimating

β

’s.

Estimating

β

1. Let

r

˜

i,t,

r

˜

m,t and

r

f,t denote historical individual security, market and risk free asset returns (respectively) over some period

t

=

1,2, . . . ,

T

.

2. The standard way to estimate a beta is to use a characteristic line regression:

˜

r

i,t

−

r

f,t

=

α

i

+

β

i

(

r

˜

m,t

−

r

f,t

) +

ε

˜

i,t 3. To estimate this, typically we would use monthly data 4. Typically use 5 years (60 months) of data.

,→ Why not use more data? (10 or 20 years) I Parameter instability

,→ Can we use weekly, daily, or intraday data? I Non-synchronous prices

Estimating

β

- Example

Here is an example of a (12 month) characteristic-line regression for GM Common stock (fromBKM, Ch. 10):

Estimating

β

- Example

This figure illustrates this still better:

Estimating

β

There are a number of Institutions that supply

β

’s:

1. Value-Lineuses the past five years (with weekly data) with the Value-Weighted NYSE as the market.

2. Merrill Lynchuses 5 years of monthly data with the S&P 500 as the market

Estimating

β

Merrill usestotalrather thanexcessreturns:

˜

r

i,t

=

a

i

+

b

i

r

˜

m,t

+

e

˜

i,t So, the

a

ihere is equal to:

a

i

=

α

i

+ (

1

−

β

i

)

r

f

,→ whererf is the average risk-free rate over the estimation period

,→ aiis not equal to zero if the CAPM is true.

Estimating

β

Note also that Merrill used adjusted

β

’s, which are equal to:

β

Ad j_i

≈

1/3

+ (

2/3

)

·

β

ˆ

i

,→ Intuition?

I Statistical Bias

I Why1/3and2/3?

I What should these numbers be based on?

Estimating

β

for new companies

In the absence of historical data, how would we estimate

β

for new companies?

Standard industry practice is to use “comparables”:

,→ Find a similar company, that is traded on an exchange and use the beta of that company.

The following approach is in the same spirit but is more powerful.

,→ Often a comparable company cannot be found.

,→ Can yield a model-predicted beta from a sample of companies with similar characteristics.

Estimating

β

Which characteristics to use?

,→ Industry ,→ Firm Size

,→ Financial Leverage ,→ Operating Leverage ,→ Growth / Value

Estimating

β

for new companies - Methodology

1. Select a sample of companies to estimate the model.

2. Estimate

β

i,t for these companies using historical return data. 3. Regress estimated betas

β

ˆ

i,t on several characteristics that can

drive betas. Example:

ˆ

β

i,t

=

a

0

+

∑

j

γ

j

INDU ST RY

j,i

+

a

1

FLEV

i,t

+

a

2

SIZE

i,t

+

a

3

OLEV

i,t

+

u

i,t

,→ INDU ST RYj,iis a dummy variable that take the value 1 if firmi

belongs in industryjand 0 otherwise.

4. Suppose you are asked to find the beta for new company XYZ in the tech industry, given XYZ’s characteristics. Your estimate will be

ˆ

Estimating

β

Betasmay change over time, thus we use short windows of data (5-years)

Possible reasons for this are 1. changes in the firm’s leverage

2. changes in the type of a firm’s operations 3. the firm acquires targets in other industries

We can userolling windowregressions to estimate a time series of

β

s:

,→ at monthtestimateβusing monthst−60throught−1

,→ at montht+1estimateβusing monthst−59throught.

Alternatively, use more sophisticated statistical models that allow for time-variation in beta.

Estimating

β

β

s can be quite volatile. Example: AT&T

Estimating

β

β

s of industries also changes over time. Example: Oil Industry

0 0.5 1 1.5 2 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 0 5 10 15 20 25 30 35 40 45 50 Oil industry market beta

The CAPM and Active Portfolio Management

To create an optimal portfolio we need to estimate the minimum variance/efficient frontier, MVE portfolio, and the best capital allocation line (CAL).

Where are the inputs coming from?

1. One way to do this is to ignore market opinion, and to estimate the E(ri)’s andσi,j’s for all assets.

I For example, one could use past empirical data.

I we have already seen that there are problems with this approach

2. An alternative approach is to “accept” market opinion and simply hold the market portfolio.

I This approach doesn’t tell you what to do if you have information that you believe has not yet been incorporated into market prices.

The CAPM and Active Portfolio Management

The approach we will instead take is to

1. Calculateβ’s for the securities we plan to hold using the methods discussed earlier .

2. Using theseβ’s, calculateσi,j’s andE(ri)’s, assuming the CAPM

holds exactly.

3. Then, incorporate our information by “perturbing” the values away from the CAPM-calculated values.

4. Finally, using these estimates and the Markowitz portfolio

optimization tools (from Lecture 2), determine our optimal portfolio weights.

Note that if

(i) We start with all of the assets in market portfolio

(ii) We use the unmodifiedσi,j’s andE(ri)’s we get from step 2 Then the weights we calculate in step D.4 will be exactly those of the market portfolio. Why??

A single index model

An alternative to estimating

σ

i,j’s is to assume that asingle index modeldescribes returns:

1. r˜i,t−rf,t=αi+βi(r˜m,t−rf) +ε˜i,t 2. for two different securitiesiand j,

cov(ε˜i,t,ε˜j,t) =0.

I The CAPM doesnotimply this.

I Is this a reasonable simplification?

To get covariances/correlations, use:

σ

i,j

=

cov

(

r

i,t

,

r

j,t

) =

β

i

β

j

σ

2m

∀

i

6

=

j

and

ρ

i,j

=

β

i

β

j

σ

2m

σ

i

σ

j For variances:

σ

i,i

=

σ

2i

=

β

i2

·

σ

2m

+

σ

2ε,i

The single index model vs the CAPM

1. The single index model is astatisticalmodel.

,→ It specifies thatallcommon movements between stocks can be captured by asingleindex.

,→ It is also often called asingle factor model. ,→ Can be generalized to multiple common factors.

,→ It is a statistical technique designed to estimate large covariance matrices.

2. The CAPM is aneconomicmodel of expected returns.

,→ It specifies that the market portfolio capturessystematicrisk. ,→ However, itdoesallow securitiesiand jto be correlated on top of

what their covariance with the market portfolio implies, for instance if they are in the same industry. The only requirement is that these correlations “wash out” as we add more stocks.

The CAPM and Active Portfolio Management

To get the market’s beliefs about expected returns, use the CAPM:

E

(

r

˜

i

)

−

r

f

=

β

i

·

(

E

(

r

˜

m

)

−

r

f

)

Remember, this is estimated by the regression

˜

r

i,t

−

r

f,t

=

α

i

+

β

i

(

r

˜

m,t

−

r

f

) +

˜

ε

i,t

,→ Given our estimatesαi,βiandE(r˜m,t−rf), if we calculate

αi+βiE(r˜m,t−rf)

it will be the same as estimatingE(r˜i,t−rf,t)using historical

averages. Why?

,→ The CAPM estimate ofE(r˜i,t−rf,t)is obtained by imposing

The CAPM and Active Portfolio Management

Then, to construct the efficient frontier based on the Single Index Model (for 100 securities) we need estimates of the following:

r

f 1 1

E

(

r

m

)

1 1

σ

2m 1 1

α

i N 100

β

i N 100

σ

2_ε,i N 100 Total

3

N

+

3

303

This is considerably smaller than the 5150 we had before. Where are the

E

(

r

i

)

?

Where are the

σ

i,j

=

Cov

(

r

˜

i

,

r

˜

j

)

?

What will the efficient portfolio weights be if we assume all

α

iare zero? (and include all assets in our analysis)

Example (cont)

Monthly data for GE, IBM, Exxon (XON), and GM: Excess Returns

mean std alpha beta

std

ε

R

2ad j

IBM 3.22% 8.44% 1.31% 1.14 7.13% 28.5% XON 1.41% 4.03% 0.42% 0.59 3.28% 33.7% GM 0.64% 7.34% -1.06% 1.02 6.14% 30.0% GE 2.26% 5.86% 0.53% 1.04 4.15% 49.9% VW-Rf 1.67% 4.02% Rf 0.36% 0.05%

,→ VW is the Value-Weighted index of all NYSE, AMEX, and NASDAQ common stocks.

,→ Rf is the (nominal) 1-month T-Bill yield, 4.394%/year (0.359%/month)

Example (cont)

For expected returns, let’s not impose the CAPM just yet. To get the correlation structure, we use:

ρ

i,j

=

β

i

β

j

σ

2m

σ

i

σ

j

∀

i

6

=

j

IBM XON GM GE IBM 1 0.32 0.30 0.39 XON 0.32 1 0.33 0.42 GM 0.30 0.33 1 0.40 GE 0.39 0.42 0.40 1

Example (cont)

Plugging the (1) “expected” returns, (2) return standard

deviations, and (3) correlation matrix into the Excel spreadsheet we get the following weights for the tangency portfolio:

Weight

IBM 29.6 %

XON 49.7 %

GM -21.4 %

GE 42.4 %

It seems unreasonable that we should hold such extreme portfolio positions.

,→ The equilibrium arguments we used in developing the CAPM suggest that the market knows something we don’t about future expected returns!

Example (cont)

Let’s use the CAPM as a way of getting around this problem. This is equivalent to setting

α

i

=

0

for all securities, or using the regression equation:

E

(

r

i

) =

r

f

+

β

i

[

E

(

r

m

)

−

r

f

]

and the past (average) return on the market to get “equilibrium” estimates of the expected returns.

Stock CAPM

E

(

r

_ie

)

IBM 1.91 %

XON 0.99 %

GM 1.70 %

Example (cont)

With this “equilibrium” set of expected returns, we now get the portfolio weights: weights IBM 13.6% XON 33.3% GM 16.4% GE 36.6%

,→ Why are the weights different?

,→ Why are these not the market weights? When would these be the actual market weights?

,→ Is this the portfolio you would want to hold, given that you were constrained to hold these four assets?

Example (cont)

However, there may be times when we think that the market is a little wrong along one or more dimensions (a very dangerous assumption!)

How can we combine our views with what the market expects? 1. First, suppose that I think that the “market” has underestimated the earnings that IBM will announce in the next month, and that IBM’s expected return is 2% higher than the market expects. 2. Also, I have no information on the other three securities that would

lead me to think that they are mispriced,

3. and I believe that the past betas, and residual std dev’s are good indicators of the relative future values.

Example (cont)

In this case, we would use the same variance and covariance inputs, but would change the expected returns. The new portfolio weights would be:

Stock

E

(

r

_ie

)

weights

IBM 3.91% 54.1%

XON 0.99% 17.7%

GM 1.70% 8.7%

GE 1.73% 19.5%

compared to the old allocation

E

(

r

e_i

)

weights

IBM 1.91% 13.6%

XON 0.99% 33.3%

GM 1.70% 16.4%

Example (cont)

Alternatively, suppose that I think the risk (

β

) of Exxon is increasing.

1. I guess that Exxon’sβwill rise from 0.59 to 0.8.

2. I also expect that Exxon’s idiosyncratic riskσεwill not change. First, I should recalculatealmosteverything using the equations:

E

(

r

i

)

=

r

f

+

β

i

[

E

(

r

m

)

−

r

f

]

σ

2i

=

β

2i

·

σ

2m

+

σ

2ε,i

ρ

i,j

=

β

i

β

j

σ

2m

σ

i

σ

j

Example (cont)

The new correlations we come up with are:

IBM XON GM GE

IBM 1.00 0.44 0.30 0.39

XON 0.44 1.00 0.45 0.57

GM 0.30 0.45 1.00 0.40

GE 0.39 0.57 0.40 1.00

as opposed to the old correlation matrix of:

IBM XON GM GE

IBM 1 0.32 0.30 0.39

XON 0.32 1 0.33 0.42

GM 0.30 0.33 1 0.40

Example (cont)

If, I believe that these are the new correlations, but that the market still believes that the past correlations represent the future (and will not discover this information over the next several months) then I would use theold expected returns, giving new portfolio weights of:

old new

E

(

r

_ie

)

weight weight IBM 1.91% 13.6% 17.0% XON 0.99% 33.3% 16.7% GM 1.70% 16.4% 20.5% GE 1.73% 36.6% 45.7%

Example (cont)

If, I believe that market knows that the

β

of Exxon is higher, and the expected return on Exxon is higher now to compensate for the increased risk:

old new

E

(

r

_ie

)

weight

E

(

r

e

)

weight IBM 1.91% 13.6% 1.91% 9.2% XON 0.99% 33.3% 1.34% 54.8% GM 1.70% 16.4% 1.70% 11.1% GE 1.73% 36.6% 1.73% 24.8%

One other alternative is that I don’t believe that the market yet knows that the risk of Exxon is higher, but will discover this in the next few months.

1. What will happen as the market finds out? 2. What should I do in this case?

Forming ’views’

Where is our information coming from?

1. It could come from public sources of information that may or may not have been impounded in prices yet.

2. It could be private, that is it could come from our own analysis of fundamentals.

Nevertheless, we may not be 100% confident that our information is accurate.

For instance, in the previous example, if I am onlysomewhat confident of my belief that the market will not adjust the price of Exxon properly, I might want to adjust the portfolio weights only part-way.

Forming ’views’ - Example

Forming ’views’ - Example

1. Market price:

P

0

=

58.95

2. Your view:

2.1 V0= (1+a)P0=65.00→a=10.26%

2.2 rCAPM= [E(P1)−V0]/V0

3. You think the actual return will be:

r

=

[

E

(

P

1

)

−

P

0

]

/

P

0

=

[

E

(

P

₁

)

−

V

₀

+

V

₀

−

P

₀

]

/

P

₀

=

r

CAPM

(

V

0

/

P

0

) + [

V

0

−

P

0

]

/

P

0

=

r

CAPM

+

a

+

r

CAPM

×

a

Forming ’views’ - Example

If you were 100% confident in your view, then you would also think that AIG’s expected return will be higher than the CAPM implied return by an amount

a

=

10.26%

.

This is a very large number and will likely lead to extreme portfolio allocations.

Suppose that you are only

10%

confident in your view, then you might use

a

=

0.1

×

10.26%

=

1.26%

.

NEXT: The Black and Litterman model gives a systematic framework that enables you to incorporate your views with what the market’s views in forming the optimal portfolio.

Black-Litterman model - Global Asset Allocation

Focus on asset allocation between country indices. Use an international version of the CAPM.

Calculate the “equilibrium” expected returns based on estimated variances and covariances. This is appropriate, since

covariances can be estimated accurately, especially using daily data, for a small number of assets.

The portfolio manager can specify any number of market “views" in the form of expected returns, and a variance (measure of uncertainty) for each of the views.

,→ If the manager holds no views, she will hold the market portfolio. ,→ If her views are high variance (low certainty), she will hold close to

the equilibrium portfolio.

,→ When her views are low variance, she will move considerably away from the market portfolio.

Black-Litterman model - Global Asset Allocation

First, let’s look at the sort of portfolio allocation we get if we use historical returns and volatilities as inputs:

Black-Litterman model - Global Asset Allocation

Black-Litterman model - Global Asset Allocation

If you use our procedures from Lecture 2 and calculate and optimal portfolio, with

σ

p

=

10.7%

, you will get portfolio weights of:

Black-Litterman model - Global Asset Allocation

CAPM Based Estimates

Black-Litterman model - Global Asset Allocation

There are N assets in the market, returns are normally distributed:

R

∼

N

(

µ

,

Σ

)

µ

is an unknown quantity. The CAPM estimate says

µ

∼

N

(

Π

,

τΣ

)

What this means is that the investor is uncertain about what the expected returns are. He will form his “best guess” or in Bayesian terminology hisposteriorbeliefs, by combining information from two sources:

,→ The first, hispriorΠ, will be based on the CAPM. The variance of his prior,τΣ, denotes how much confidence the investor has on the CAPM. High values ofτmeans that he attaches less weight to the CAPM.

Black-Litterman model - Global Asset Allocation

Suppose that we believe that Germany will have an expected return of 4%

Black-Litterman allows you to express this view as

µ

ger

=

q

+

ε

,

ε

∼

N

(

0,

ω

)

Here,

q

=

4%

is your belief about Germany’s expected return. The variable

ε

allows for a margin of error in your view. Your confidence in your view is represented by

ω

, i.e. the variance in your margin of error.

Black-Litterman model - Global Asset Allocation

The Black-Litterman model also allows you to form relative views.

Suppose that we believe that Germany will outperform France by 5%

Black-Litterman allows you to express this view as

Pµ

=

Q

+

ε

,

ε

∼

N

(

0,

Ω

)

Here,

P

is a matrix, for instance if there are only two countries and you only form a relative view, it will equal

P

=

1

−

1

0

0

Black-Litterman model - Global Asset Allocation

Given our views and the CAPM prior, our “best guess” about what expected returns are is

µ

is going to be a weighted average of the CAPM expected returns,

Π

and our views,

Q

.

µ

= [(

τΣ

)

−1

+

P

0

Ω

−1

P

]

−1

[(

τΣ

)

−1

Π

+

P

0

Ω

−1

Q

]

In the case of one asset with prior

π

and one view,

q

, it simplifies to

µ

=

π

1/

(

σ

2

τ

)

1/

ω

+

1/

(

σ

2

τ

)

+

q

1/

ω

1/

ω

+

1/

(

σ

2

τ

)

Forming posterior beliefs - Example

Suppose that you would like to for your best guess about GE’s expected excess return going forward.

,→ GE has a beta of 1.2 and a standard deviation of 0.33. Assuming the equity premium is 6%, this gives you (in excess of the risk-free rate)

µcapm_GE =1.2×6%=7.2% and var(µcapm_GE ) =τ×0.332=0.1×τ ,→ You estimated the historical average return of GE to be 8.3% over

the last 10 years. This gives you another estimate or “view”:

µhist_GE=8.3% and var(µhist_GE) =

_0.33 √

10 2

Forming posterior beliefs - Example

Your best guess will be:

µ

GE

=

1 var(µcapm_GE )

×

µ

capm GE

+

1 var(µhist GE)

×

µ

hist_GE 1 var(µcapm_GE )

+

1 var(µhist GE)

=

1 0.1×τ

×

7.2%

+

1 0.01

×

8.3%

1 0.1×τ

+

1 0.01

It is a weighted average of your prior (the CAPM) and your view (the historical average return).

The relative weights depend on the precision (the inverse of the variance) of the signals.

Forming posterior beliefs - Example

Your posterior belief as a function of

τ

0 0.5 1 1.5 2 2.5 3 0.072 0.074 0.076 0.078 0.08 0.082 0.084 τ µGE

Black-Litterman model - Global Asset Allocation

−5 −4 −3 −2 −1 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 prior (CAPM) ’best guess’ view µ bar

Black-Litterman model - Global Asset Allocation

Finally, the optimal portfolio weights will be:

w

∗

=

w

MKT

+

P

0

Λ

where

P

are the investor’s “view” portfolios, and

Λ

is a complicated set of weights. These weights have the following properties:

1. The higher the expected return on a view,q, the higher the weight attached to that view,λ.

2. The higher the variance of a view,ω, the lower the absolute value of the weight attached to the view.

If you hold no view on an asset, then your optimal allocation will be the market weight.

Black-Litterman model - Global Asset Allocation

Suppose that we believe that Germany will outperform France and UK.

Black-Litterman model - Global Asset Allocation

Black-Litterman model - Global Asset Allocation

Our view is that Germany will outperform France and UK. However, our best guess,

µ

, will be shifted forallcountries.

Black-Litterman model - Global Asset Allocation

The Black-Litterman model internalizes the fact that assets are correlated. Your views about Germany vs France and UK translate intoimplicitviews about other countries.

Black-Litterman model - Global Asset Allocation

Black-Litterman model - Global Asset Allocation

weights depend

How bullish is the view, i.e. (

Q

)

Black-Litterman model vs Markowitz

Black Litterman

1. The optimal portfolio equals the (CAPM) market portfolio plus a weighted sum of the portfolios about which the investor has views. 2. An unconstrained investor will invest first in the market portfolio,

then in the portfolios about which views are expressed.

3. The investor will never deviate from the market weights on assets about which no views are held.

4. This means that an investor does not need to hold views about each and every asset!

Markowitz

1. Need to estimate vector of expected returns for all assets. 2. Estimation error often leads to unrealistic portfolio positions 3. If we change the expected return for one asset, this will change

Summary

In order to use the CAPM in our mean-variance analysis, we need estimates of

β

. Regression analysis yields that.

The CAPM assumes that all investors hold the same views so all of them hold the market-portfolio.

The Black-Litterman model is one way to bring the information from the CAPM in our asset allocation decision.